Sketch the graph of a function that has the given properties. Identify any critical points, singular points, local maxima and minima, and inflection points. Assume that f is continuous and its derivatives exist everywhere unless the contrary is implied or explicitly stated. $f(-1)=0, f(0)=2, f(1)=1, f(2)=0, f(3)=1$, $\lim _{x \rightarrow \pm \infty}(f(x)+1-x)=0, f^{\prime}(x)>0 \text { on }(-\infty,-1)$, $(-1,0) \text { and }(2, \infty), f^{\prime}(x)<0 \text { on }(0,2)$, $\lim _{x \rightarrow-1} f^{\prime}(x)=\infty, f^{\prime \prime}(x)>0 \text { on }(-\infty,-1)$ and on (1, 3), and f"(x) < 0 on (-1, 1) and on $(3, \infty)$.

In a series RLC circuit, $R=X_{C}=X_{L}=40 \Omega$ for a particular driving frequency. (a) This circuit is (1) inductive, (2) capacitive, (3) in resonance. Explain your reasoning. (b) If the driving frequency is doubled, what will be the impedance of the circuit?

Sketch the graphs of the given functions, making use of any suitable information you can obtain from the function and its first and second derivatives. $y=\frac{x^{3}}{x^{2}-1}$

Find the area of the region. Two petals of $r=\sin 8 \theta$

Suppose that the trapezoidal rule is used to estimate the value of $\int_{0}^{0.2} \int_{-0.1}^{0.1} e^{x^{2}+2 y} d y d x$. Determine the smallest value of n so that the resulting approximation $T_{n, n}$ is accurate to within $10^{-4}$ of the actual value of the integral.

Find AB, if possible.

$$A=\left[\begin{array}{rr} {-1} & {3} \\ {4} & {-5} \\ {0} & {2} \end{array}\right], B=\left[\begin{array}{ll} {1} & {2} \\ {0} & {7} \end{array}\right]$$

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a) Graph $y=\tan \theta$ for $-2 \pi \leq \theta \leq 0$ b) State the equations of the asymptotes for the graph in part a).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The set of points $\left\{(x, y, z) | x^{2}+y^{2}=1\right\}$ is a circle.

Let $f(x)=\cos x$. Show that finding $f^{\prime}(0)$ is the same as finding $\lim _{x \rightarrow 0} \frac{\cos x-1}{x}$.

In your own words, explain why $0 \cdot \infty \neq 0$.

Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$. If the Mean Value Theorem cannot be applied, explain why not. $f(x)=x^{6}$, [-1, 1]

Determine the equation of the transformed image after the transformations described are applied to the given graph. a) The graph of $y=2 \log _{5} x-7$ is reflected in the x-axis and translated 6 units up. b) The graph of y = log (6(x - 3)) is stretched horizontally about the y-axis by a factor of 3 and translated 9 units left.

Write a proof for each limit using the $\varepsilon-\delta$ definition of a limit. $\lim _{x \rightarrow 3}(4 x)=12$

For each of the following situations, decide what sampling method you would use. Provide an explanation of why you selected a particular method of sampling. a. Alarge automotive companywants to upgrade the software on its notebook computers. A survey of 1,500 employees will request information concerning frequently used software applications such as spreadsheets,word processing, e-mail, Internet access, statistical data processing, and so on.Alist of employees with their job categories is available. b. A hospital is interested in what types of patients make use of their emergency room facilities. It is decided to sample 10% of all patients arriving at the emergency room for the next month and record their demographic information along with type of service required, the amount of time patient waits prior to examination, and the amount of time needed for the doctor to assess the patient’s problem.